Center manifold reduction for large populations of globally coupled phase oscillators

Chiba, Hayato; Nishikawa, Isao

doi:10.1063/1.3647317

Cited by 55 publications

(79 citation statements)

References 19 publications

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“…[2][3][4][8][9][10][11][12] In general, the phase oscillator model with uniform natural frequency coincides with the classical XY model at zero temperature, if the interactions in both models are the same. We are now studying the phase oscillator network model with the Mexican-hat type interaction and clarifying the resemblance of both models.…”

Section: Numerical Resultsmentioning

confidence: 99%

Analysis of XY Model with Mexican-Hat Interaction on a Circle –Derivation of Saddle Point Equations and Study of Bifurcation Structure–

2012

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In our previous study, we investigated a classical XY model on a circle by adopting the Mexican-hat type interaction, which is composed of uniform and location-dependent interactions. We solved the saddle point equations numerically and found three nontrivial solutions. In this study, we determined the phases of complex order parameters and derived the saddle point equations for stable and unstable nontrivial solutions and the formula of boundaries of bistable regions analytically. We performed Markov Chain Monte Carlo simulations and confirmed that the numerical and theoretical results agree well.KEYWORDS: XY model, Mexican-hat interaction, saddle point equations, bistability §1. IntroductionOver these past years, we have been studying the synchronization -desynchronization phase transition of oscillator networks. 1 In particular, we have studied the phase oscillator network [2][3][4] with the Mexican-hat type interaction on a circle. This type of interaction was introduced to model the feature extraction cells in neurosciences 5,6 and to express effects of excitation of nearby neurons and inhibition of distant neurons.In the course of the analysis of the phase oscillator network, it turned out that information on the phases of complex order parameters is necessary. Therefore, we studied the XY model on a circle with the same interaction as the phase oscillator network, because both models coincide with each other under some conditions. In the XY model, we found three nontrivial solutions of the saddle point equations (SPEs), the uniform (U), spinning (S), and pendulum (Pn) solutions. 7 We confirmed the agreement between the theoretical and numerical results, and drew phase diagrams by performing numerical simulations.In this study, we theoretically determined the phases of complex order parameters that enabled us to derive the self-consistent equations (SCEs) of the amplitudes of complex order parameters in the phase oscillator network. We derived the SPEs of the amplitudes of complex order param- *

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Section: Numerical Resultsmentioning

confidence: 99%

Analysis of XY Model with Mexican-Hat Interaction on a Circle –Derivation of Saddle Point Equations and Study of Bifurcation Structure–

2012

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“…, which corresponds to the second law of thermodynamics. Furthermore, from the identity (27), in the nearly quasi-static regime η → 0, we have…”

Section: Stochastic Thermodynamicsmentioning

confidence: 99%

Collective dynamics from stochastic thermodynamics

Sasa

2015

New J. Phys.

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From a viewpoint of stochastic thermodynamics, we derive equations that describe the collective dynamics near the order-disorder transition in the globally coupled XY model and near the synchronization-desynchronization transition in the Kuramoto model. A new way of thinking is to interpret the deterministic time evolution of a macroscopic variable as an external operation to a thermodynamic system. We then find that the irreversible work determines the equation for the collective dynamics. When analyzing the Kuramoto model, we employ a generalized concept of irreversible work which originates from a non-equilibrium identity associated with steady state thermodynamics.

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“…The last decade has seen considerable research on coupled systems on networks (CSNs) due to their extensive applications in physics, epidemiology, and neural networks . Although it is well known that stability analysis of CSNs is a prerequisite and essential work, examining the stability of CSNs is generally a challenging task since the connections between stable and isolated units do not guarantee the stability of CSNs and may even introduce instability.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic stability in probability for discrete‐time stochastic coupled systems on networks with multiple dispersal

Wang

Hong

2017

Intl J Robust & Nonlinear

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Summary In this paper, we consider the asymptotic stability in probability for discrete‐time stochastic coupled systems on networks with multiple dispersal (DSCSM). We begin with modeling a DSCSM on multiple digraphs and consequently construct a global Lyapunov function based on the topological structure of multiple digraphs. Using the Lyapunov method combined with the graph theory and the supermartingale convergence theorem, several stability criteria for DSCSM are derived. In what follows, the given results are utilized to analyze a stochastic coupled oscillator model. Finally, 2 numerical examples are also given to demonstrate the feasibility of the proposed results.

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Center manifold reduction for large populations of globally coupled phase oscillators

Cited by 55 publications

References 19 publications

Analysis of XY Model with Mexican-Hat Interaction on a Circle –Derivation of Saddle Point Equations and Study of Bifurcation Structure–

Analysis of XY Model with Mexican-Hat Interaction on a Circle –Derivation of Saddle Point Equations and Study of Bifurcation Structure–

Collective dynamics from stochastic thermodynamics

Asymptotic stability in probability for discrete‐time stochastic coupled systems on networks with multiple dispersal

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